Asignment Week 1
Shawn Schneidereit
2.11.2020
note that the .Rmd can be found on my personal repo for the corse
Constraining the effects of topography and climate on climate change sensitivity of glaciers in Tibet
The Tibetan Plateau region is home to the greatest amount of non-polar glaciers, which are also highly sensitive to climate change and has recently been experiencing the greatest rates of mass losses. These glaciers are primary freshwater sources to major Asian rivers, with increased rates of melting increasing the risks of floods and other geohazards to some of the world’s most densely populated regions. Yet the geophysical and atmospheric conditions driving changes in Tibetan plateau glacier mass balance remain relatively understudied, with uncertainties in which variables can best predict glacier mass change. In this analysis I will use three different modelling approaches to assess which topographical and climatological drives are most important in explaining changes in glacier equilibrium line altitude (ELA) and whether these drives can be used to explain glacier ELA changes since the little ice age. The first approach uses a Mixed effects model with random effects, where all variables where chosen based on a priori knowledge gained from the readings and an exploratory correlation plot. As the exploratory correlation plot showed that many variables displayed high levels of collinearity/multicollinearity, I decided to create a model using principal components from a PCA, to preserve all available variables, while not violating any assumptions of variable collinearity and codependence. Finally, I decided to use an automatic variable selection method (MUMIn), where I included the maximum number of variables possible, within the computational limits of my PC. While I believe that xxx method best models which variables influence ELA and dELA, because xxx, I synthesize the results all three models to judge xxx.
In this analysis I attempted to justify all modelling structure and variable selection decisions made. As I quite unfamiliar with climatic modelling and glacier dynamics, some of these decisions may be sub-optimal. Yet, these decisions were made to the best of my abilities, without going over the top on background research and overall, my models should have a coherent structure and sufficient workflow documentation. In none of the preliminary readings, did I encountered the temperature or precipitation in a specific month singled out to be of particular importance for glacier ELA. Therefore, I decided to exclude all monthly climatic variables and only focus on the aggregated factors. Along with the provided variables, I also included a new variable called temperature variability, as my preliminary readings highlighted its potential importance (Brun, et. al., 2017) and I was able to compute it with the provided dataset.
Multiple sources confirm that overall there is a lack of knowledge on climatic data and their influence on the TP, due to a lack of Permanente meteorological weather stations in the TP, with the influence of Monsoon being aknoweleged (Maussion et. al., 2014).
1.Data import and wrangeling
glacier <- read_csv("~/Documents/humbolt/quantitative_methods/assignments/glacier_sensitivity.csv") %>%
na.exclude ## Parsed with column specification:
## cols(
## .default = col_double(),
## orientation = col_character()
## )## See spec(...) for full column specifications.glacier_scaled <- glacier %>%
# adding a new variable for temperature variability; a factor mentioned to be important in multiple papers
mutate(T_variability = ((T_MAX_mean_monsoon + T_MAX_mean_not_monsoon)-(T_MIN_mean_monsoon + T_MIN_mean_not_monsoon) /2),
# scaling all varibales used in modelling
length = scale(length, center = T, scale = T),
area = scale(area, center = T, scale = T),
P_snow = scale(P_snow, center = T, scale = T),
P_year = scale(P_year, center = T, scale = T),
P_monsoon = scale(P_monsoon, center = T, scale = T),
P_not_monsoon = scale(P_not_monsoon, center = T, scale = T),
T_MIN_mean_monsoon = scale(T_MIN_mean_monsoon, center = T, scale = T),
T_MIN_mean_not_monsoon = scale(T_MIN_mean_not_monsoon, center = T, scale = T),
T_MAX_mean_monsoon = scale(T_MAX_mean_monsoon, center = T, scale = T),
T_MAX_mean_not_monsoon = scale(T_MAX_mean_not_monsoon, center = T, scale = T),
T_variability = scale(T_variability, center = T, scale = T),
T_mean_mea.yr = scale(T_mean_mea.yr, center = T, scale = T),
T_mean_monsoon = scale(T_mean_monsoon, center = T, scale = T),
T_mean_not_monsoon = scale(T_mean_not_monsoon, center = T, scale = T),
Slope_min = scale(Slope_min, center = T, scale = T),
Slope_max = scale(Slope_max, center = T, scale = T),
Slope_mean = scale(Slope_mean, center = T, scale = T),
Elev_min = scale(Elev_min, center = T, scale = T),
Elev_max = scale(Elev_max, center = T, scale = T),
Elev_mean = scale(Elev_mean, center = T, scale = T))
#ggplot(data=glacier, aes_string(x = id, y = "")) + geom_line()
colnames(glacier) ## [1] "id" "ELA" "dELA"
## [4] "summit" "debris_cov" "morph_type"
## [7] "orientation" "length" "area"
## [10] "P_01_mean" "P_02_mean" "P_03_mean"
## [13] "P_04_mean" "P_05mean" "P_06_mean"
## [16] "P_07_mean" "P_08_mean" "P_09_mean"
## [19] "P_10_mean" "P_11_mean" "P_12_mean"
## [22] "P_snow" "P_year" "P_monsoon"
## [25] "P_not_monsoon" "T_MIN_mean.1" "T_MIN_mean.2"
## [28] "T_MIN_mean.3" "T_MIN_mean.4" "T_MIN_mean.5"
## [31] "T_MIN_mean.6" "T_MIN_mean.7" "T_MIN_mean.8"
## [34] "T_MIN_mean.9" "T_MIN_mean.10" "T_MIN_mean.11"
## [37] "T_MIN_mean.12" "T_MIN_mean_monsoon" "T_MIN_mean_not_monsoon"
## [40] "T_MAX_mean.1" "T_MAX_mean.2" "T_MAX_mean.3"
## [43] "T_MAX_mean.4" "T_MAX_mean.5" "T_MAX_mean.6"
## [46] "T_MAX_mean.7" "T_MAX_mean.8" "T_MAX_mean.9"
## [49] "T_MAX_mean.10" "T_MAX_mean.11" "T_MAX_mean.12"
## [52] "T_MAX_mean_monsoon" "T_MAX_mean_not_monsoon" "T_mean_mea.1"
## [55] "T_mean_mea.2" "T_mean_mea.3" "T_mean_mea.4"
## [58] "T_mean_mea.5" "T_mean_mea.6" "T_mean_mea.7"
## [61] "T_mean_mea.8" "T_mean_mea.9" "T_mean_mea.10"
## [64] "T_mean_mea.11" "T_mean_mea.12" "T_mean_mea.yr"
## [67] "T_mean_monsoon" "T_mean_not_monsoon" "Slope_min"
## [70] "Slope_max" "Slope_mean" "Elev_min"
## [73] "Elev_max" "Elev_mean"#(corr_plot <- ggpairs(data = glacier, columns = c(2:9,22:25,38:39,52:53,66:74)))
#corplot?
#corrplot.mixed(cor1[,2:9,22:25,38:39,52:53,66:74], lower.col = "black" , number.cex = .7)
# Here a number of strong correlations between variables can be seen. Notabale is this Summit and Max elevation seem to be the same variable and that xxxxQ1. Which are the most important topographical and climatological drivers of glacier equilibrium line altitude (ELA)?
Mixed effects model with random effects
In this first section, I chose to create a Mixed effects model with random effects (lme4 package), where all variables where chosen based on a priori knowelge gained from the readings and the exploratory correlation plot. The methodoological advantage of this approach is that through including the random effects, correlations between data coming from specific morphology types and geographical orientations can be accounted for, by treating them as a grouping factor. While this prevents us from explicitly assessing the impacts of morphology and geographic orientation, it still allows us to see how they influence the observed patterns in other varibales. In the final section of this analysis (the automated variable selection) morphology and geographic orientation are left as fixed effects, so that I have them once as a fixed and once as a random effect, to allow for compairson between the two secarios. As the Distribution of ELA had a slight negative skew, I also tried running a GLM with a inverse gaussian distribution (link = “1/mu^2”), but encoured the error that the “PIRLS step-halvings failed to reduce deviance in pwrssUpdate”, to which I did not find a solution.
hist(glacier_scaled$ELA) # there is a negative skew to the data, but overall a gaussian distibution looks appropriate
m_lme4 <- lmer(ELA ~ debris_cov + length + area + P_snow + P_monsoon + P_not_monsoon + T_mean_mea.yr + T_variability + Slope_mean + Elev_max + Elev_mean + (1|morph_type) + (1|orientation),
data = glacier_scaled)
summary(m_lme4)## Linear mixed model fit by REML ['lmerMod']
## Formula:
## ELA ~ debris_cov + length + area + P_snow + P_monsoon + P_not_monsoon +
## T_mean_mea.yr + T_variability + Slope_mean + Elev_max + Elev_mean +
## (1 | morph_type) + (1 | orientation)
## Data: glacier_scaled
##
## REML criterion at convergence: 9555.8
##
## Scaled residuals:
## Min 1Q Median 3Q Max
## -6.1406 -0.4238 0.0367 0.4765 6.2845
##
## Random effects:
## Groups Name Variance Std.Dev.
## orientation (Intercept) 129.3 11.37
## morph_type (Intercept) 469.1 21.66
## Residual 4819.7 69.42
## Number of obs: 849, groups: orientation, 8; morph_type, 6
##
## Fixed effects:
## Estimate Std. Error t value
## (Intercept) 5098.662 12.040 423.471
## debris_cov 2.774 11.248 0.247
## length -23.132 6.984 -3.312
## area -21.740 6.526 -3.332
## P_snow 20.095 4.940 4.068
## P_monsoon 8.067 4.476 1.802
## P_not_monsoon -33.825 7.117 -4.753
## T_mean_mea.yr -101.071 10.953 -9.228
## T_variability 59.367 7.939 7.478
## Slope_mean -6.581 3.339 -1.971
## Elev_max 8.238 5.691 1.448
## Elev_mean 141.943 7.940 17.876
##
## Correlation of Fixed Effects:
## (Intr) dbrs_c length area P_snow P_mnsn P_nt_m T_mn_. T_vrbl
## debris_cov -0.096
## length -0.059 -0.044
## area -0.031 -0.060 -0.722
## P_snow 0.022 -0.004 -0.039 0.069
## P_monsoon -0.006 0.039 -0.163 0.077 0.016
## P_not_monsn -0.026 0.023 0.126 -0.127 -0.713 -0.546
## T_mean_m.yr 0.024 -0.081 0.021 -0.085 0.026 -0.047 -0.138
## T_variablty 0.008 -0.013 -0.058 0.124 0.256 0.115 -0.116 -0.689
## Slope_mean 0.059 0.005 0.095 0.177 0.066 -0.061 0.025 -0.384 0.177
## Elev_max -0.012 -0.131 -0.319 -0.191 -0.028 0.010 0.090 -0.091 0.165
## Elev_mean 0.031 -0.016 0.160 0.041 -0.109 -0.094 0.132 0.612 -0.105
## Slp_mn Elv_mx
## debris_cov
## length
## area
## P_snow
## P_monsoon
## P_not_monsn
## T_mean_m.yr
## T_variablty
## Slope_mean
## Elev_max -0.477
## Elev_mean -0.067 -0.371# R squared
r.squaredGLMM(m_lme4)## Warning: 'r.squaredGLMM' now calculates a revised statistic. See the help page.## R2m R2c
## [1,] 0.8996507 0.9107332# The R²(C) is 90.78 %, where the C represents the Conditional R_GLMM² and can be interpreted as a variance explained by the entire model, including both fixed and random effects
# check out variable co-lineararity
plot(m_lme4)
# the residual vs fitted plot shows a mostly flat and even distribution of points, with greater diviances seen at the lower spectrum of points, likely due to the negative skew described above
qqnorm(resid(m_lme4))
qqline(resid(m_lme4))
# When checking the QQ-plot it is seen that there is a left tail at lower bonds and a right tail at the upper bonds. Following an ad hoc transformation of the dependant variable ELA, this observed trend only showed a very marginal improvment, so I decided to keep ELA in its untransformed state.
dwplot(m_lme4) +
geom_vline(xintercept = 0, colour = "grey60", linetype = 2)
hist(glacier_scaled$dELA) # there is a stronger skew to the data. While again an inverse Gaussian distribution seems approaprite, this was also not possible. Instead I compaired a the model with a normal Gaussian and a Possion distribution and found that the Possion had a better distribution of residuals.
m_lme4_d <- glmer(dELA ~ debris_cov + length + area + P_snow + P_monsoon + P_not_monsoon + T_mean_mea.yr + T_variability + Slope_mean + Elev_max + Elev_mean + (1|morph_type) + (1|orientation),
family = poisson ,data = glacier_scaled)
summary(m_lme4_d)## Generalized linear mixed model fit by maximum likelihood (Laplace
## Approximation) [glmerMod]
## Family: poisson ( log )
## Formula:
## dELA ~ debris_cov + length + area + P_snow + P_monsoon + P_not_monsoon +
## T_mean_mea.yr + T_variability + Slope_mean + Elev_max + Elev_mean +
## (1 | morph_type) + (1 | orientation)
## Data: glacier_scaled
##
## AIC BIC logLik deviance df.resid
## 26667.4 26733.8 -13319.7 26639.4 835
##
## Scaled residuals:
## Min 1Q Median 3Q Max
## -9.4921 -3.6701 -0.7578 2.7556 30.8457
##
## Random effects:
## Groups Name Variance Std.Dev.
## orientation (Intercept) 0.01299 0.1140
## morph_type (Intercept) 0.68000 0.8246
## Number of obs: 849, groups: orientation, 8; morph_type, 6
##
## Fixed effects:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) 4.297853 0.342806 12.537 < 2e-16 ***
## debris_cov -0.287964 0.016273 -17.696 < 2e-16 ***
## length 0.022226 0.009986 2.226 0.026033 *
## area -0.004882 0.009229 -0.529 0.596816
## P_snow 0.034999 0.007133 4.907 9.27e-07 ***
## P_monsoon 0.047426 0.006502 7.294 3.01e-13 ***
## P_not_monsoon -0.104105 0.010147 -10.260 < 2e-16 ***
## T_mean_mea.yr 0.230202 0.014964 15.383 < 2e-16 ***
## T_variability -0.022872 0.010742 -2.129 0.033236 *
## Slope_mean 0.091553 0.004734 19.339 < 2e-16 ***
## Elev_max 0.059421 0.007670 7.747 9.39e-15 ***
## Elev_mean 0.038329 0.010282 3.728 0.000193 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Correlation of Fixed Effects:
## (Intr) dbrs_c length area P_snow P_mnsn P_nt_m T_mn_. T_vrbl
## debris_cov -0.002
## length -0.007 -0.040
## area -0.002 -0.071 -0.706
## P_snow 0.004 -0.015 -0.039 0.075
## P_monsoon 0.002 0.052 -0.184 0.067 0.010
## P_not_monsn -0.005 0.028 0.151 -0.135 -0.718 -0.552
## T_mean_m.yr 0.001 -0.093 -0.020 -0.080 0.011 -0.036 -0.105
## T_variablty 0.000 -0.017 -0.047 0.125 0.254 0.168 -0.175 -0.736
## Slope_mean 0.003 0.005 0.119 0.167 0.078 -0.101 0.023 -0.419 0.215
## Elev_max -0.001 -0.105 -0.322 -0.194 -0.034 0.047 0.063 -0.049 0.129
## Elev_mean 0.000 -0.052 0.134 0.040 -0.149 -0.039 0.145 0.605 -0.181
## Slp_mn Elv_mx
## debris_cov
## length
## area
## P_snow
## P_monsoon
## P_not_monsn
## T_mean_m.yr
## T_variablty
## Slope_mean
## Elev_max -0.482
## Elev_mean -0.097 -0.359# R squared (conditional)
r.squaredGLMM(m_lme4_d)## Warning: The null model is correct only if all variables used by the original
## model remain unchanged.## R2m R2c
## delta 0.04899499 0.9864700
## lognormal 0.04899829 0.9865363
## trigamma 0.04899167 0.9864031# The R²(C) is 98.65 %, where the C represents the Conditional R_GLMM² and can be interpreted as a variance explained by the entire model, including both fixed and random effects
# check out variable co-lineararity
plot(m_lme4_d)
# the residual vs fitted plot shows a mostly flat and even distribution of points, with some outlying point at intermidiate values
qqnorm(resid(m_lme4_d))
qqline(resid(m_lme4_d))
# When checking the QQ-plot it is seen that there is slight tail at the upper bound, but overall this is a significant improvement over the gaussian distribution.
dwplot(m_lme4_d) +
geom_vline(xintercept = 0, colour = "grey60", linetype = 2)
Bayesian Models
The default prior values assumed in a MCMCglmm are posterior distribution with very large variance values for the fixed effects and a flat (weakly informative) prior. The variances for the random effects are assumed to have inverse-Wishart priors with a very low value for nu (weakly informative). I decided to not dealve deeper into assigning my own priors, due to me having i) little prior knowlege the dyamics / expected values and distributions of variables and ii) a shallow knowelge of the mathimatical underpinnings and practical inplimentation methods for assigning more informative priors
m_bays <- MCMCglmm(ELA ~ debris_cov + length + area + P_snow + P_monsoon + P_not_monsoon + T_mean_mea.yr + T_variability + Slope_mean + Elev_max + Elev_mean, random = ~ morph_type + orientation,
data = glacier_scaled) ## Warning: Unknown or uninitialised column: `family`.## Warning: Setting row names on a tibble is deprecated.##
## MCMC iteration = 0
##
## MCMC iteration = 1000
##
## MCMC iteration = 2000
##
## MCMC iteration = 3000
##
## MCMC iteration = 4000
##
## MCMC iteration = 5000
##
## MCMC iteration = 6000
##
## MCMC iteration = 7000
##
## MCMC iteration = 8000
##
## MCMC iteration = 9000
##
## MCMC iteration = 10000
##
## MCMC iteration = 11000
##
## MCMC iteration = 12000
##
## MCMC iteration = 13000summary(m_bays)##
## Iterations = 3001:12991
## Thinning interval = 10
## Sample size = 1000
##
## DIC: 9643.695
##
## G-structure: ~morph_type
##
## post.mean l-95% CI u-95% CI eff.samp
## morph_type 683.3 4.404e-10 2444 782.6
##
## ~orientation
##
## post.mean l-95% CI u-95% CI eff.samp
## orientation 87.29 9.069e-17 372.6 68.91
##
## R-structure: ~units
##
## post.mean l-95% CI u-95% CI eff.samp
## units 4888 4421 5359 1000
##
## Location effects: ELA ~ debris_cov + length + area + P_snow + P_monsoon + P_not_monsoon + T_mean_mea.yr + T_variability + Slope_mean + Elev_max + Elev_mean
##
## post.mean l-95% CI u-95% CI eff.samp pMCMC
## (Intercept) 5098.6576 5069.8884 5120.5252 592.7 <0.001 ***
## debris_cov 5.0588 -18.3519 26.6213 1000.0 0.660
## length -24.6571 -38.9809 -11.2936 793.4 0.002 **
## area -22.2670 -35.0316 -9.6595 820.9 0.002 **
## P_snow 19.4200 10.4374 30.2242 1000.0 0.002 **
## P_monsoon 9.7579 -0.0156 18.9324 129.4 0.044 *
## P_not_monsoon -33.7983 -47.2544 -19.2011 1000.0 <0.001 ***
## T_mean_mea.yr -95.0648 -119.8795 -72.2362 101.7 <0.001 ***
## T_variability 56.3455 38.5928 71.1691 403.8 <0.001 ***
## Slope_mean -7.3941 -13.2619 -0.3803 572.7 0.030 *
## Elev_max 9.0243 -1.9306 19.6684 682.6 0.126
## Elev_mean 145.1785 128.1464 162.3089 231.1 <0.001 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1# R squared
#r.squaredGLMM(m_bays)
# Unfortunatly I was not able to figure out access using the r2 with a predefined function, or an official way to cumpute it. Therefore For me bayesian models I do not include r2 values.
# postirior distribution of random effects
par(mfrow = c(1,2))
hist(mcmc(m_bays$VCV)[,"morph_type"])
hist(mcmc(m_bays$VCV)[,"orientation"])
# The variance can not be equal to zero, but as the mean value is up afainst zero, this can be interpreted as the effect not being of great significance. As the spread of the histograms are reletively narrow, it can be assumed that the distribution is relatively accurate.
# Assesing model convergence through plotting trace and density est. for the intercept
plot(m_bays$Sol)
# The trace can be interpreted as the a psudo timeseries of what the model was doing in each of its iterations. As all traces have the fuzzy caterpillar appearance, it can be assumed there is adequate mixing with overall convergence being met. The density plot shows the postierer distribution of each the model predicted for each variable in each itteration. If the distribution crosses zero, the variable can be assumed to be non-significant and the spread of the distribution convays the overall accuracy of the predicted variable. m_bays_d <- MCMCglmm(dELA ~ debris_cov + length + area + P_snow + P_monsoon + P_not_monsoon + T_mean_mea.yr + T_variability + Slope_mean + Elev_max + Elev_mean, random = ~ morph_type + orientation, data = glacier_scaled) ## Warning: Unknown or uninitialised column: `family`.## Warning: Setting row names on a tibble is deprecated.##
## MCMC iteration = 0
##
## MCMC iteration = 1000
##
## MCMC iteration = 2000
##
## MCMC iteration = 3000
##
## MCMC iteration = 4000
##
## MCMC iteration = 5000
##
## MCMC iteration = 6000
##
## MCMC iteration = 7000
##
## MCMC iteration = 8000
##
## MCMC iteration = 9000
##
## MCMC iteration = 10000
##
## MCMC iteration = 11000
##
## MCMC iteration = 12000
##
## MCMC iteration = 13000summary(m_bays_d)##
## Iterations = 3001:12991
## Thinning interval = 10
## Sample size = 1000
##
## DIC: 9190.6
##
## G-structure: ~morph_type
##
## post.mean l-95% CI u-95% CI eff.samp
## morph_type 1.69 4.002e-17 2.237 170
##
## ~orientation
##
## post.mean l-95% CI u-95% CI eff.samp
## orientation 92.44 2.715e-05 303.3 756.5
##
## R-structure: ~units
##
## post.mean l-95% CI u-95% CI eff.samp
## units 2882 2600 3149 1069
##
## Location effects: dELA ~ debris_cov + length + area + P_snow + P_monsoon + P_not_monsoon + T_mean_mea.yr + T_variability + Slope_mean + Elev_max + Elev_mean
##
## post.mean l-95% CI u-95% CI eff.samp pMCMC
## (Intercept) 103.214849 95.224733 111.413266 1000.0 <0.001 ***
## debris_cov -28.829322 -45.415207 -12.708384 896.3 <0.001 ***
## length 0.063689 -10.531015 9.972741 1000.0 0.982
## area -1.241001 -10.322853 9.789121 789.7 0.796
## P_snow 3.433873 -4.232694 11.390703 1000.0 0.386
## P_monsoon 8.202609 1.241133 15.440933 1000.0 0.024 *
## P_not_monsoon -11.828262 -21.612919 0.005317 1000.0 0.038 *
## T_mean_mea.yr 33.471841 16.216489 50.786846 693.8 <0.001 ***
## T_variability -4.046931 -17.166422 8.125265 805.9 0.502
## Slope_mean 7.652471 2.496754 12.393238 824.1 0.004 **
## Elev_max 8.758933 1.015300 17.297337 1000.0 0.034 *
## Elev_mean 10.409804 -1.612909 23.000356 870.5 0.102
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1# R squared
#r.squaredGLMM(m_bays_d)
# Unfortunatly I was not able to figure out access using the r2 with a predefined function, or an official way to cumpute it. Therefore For me bayesian models I do not include r2 values.
# postirior distribution of random effects
par(mfrow = c(1,2))
hist(mcmc(m_bays_d$VCV)[,"morph_type"])
hist(mcmc(m_bays_d$VCV)[,"orientation"])
# The variance can not be equal to zero, but as the mean value is up afainst zero, this can be interpreted as the effect not being of great significance. As the spread of the histograms are reletively narrow, it can be assumed that the distribution is relatively accurate.
# Assesing model convergence through plotting trace and density est. for the intercept
plot(m_bays_d$Sol)
# The trace can be interpreted as the a psudo timeseries of what the model was doing in each of its iterations. As all traces have the fuzzy caterpillar appearance, it can be assumed there is adequate mixing with overall convergence being met. The density plot shows the postierer distribution of each the model predicted for each variable in each itteration. If the distribution crosses zero, the variable can be assumed to be non-significant and the spread of the distribution convays the overall accuracy of the predicted variable. Q1b PCA
# pca # need to redo analysis/description of pca plots
pca <- glacier_scaled[c(4:9,22:25,38:39,52:53,66:75)] %>%
dplyr::select(-orientation) # need to remove as it is a character
str(pca)## tibble [849 × 23] (S3: tbl_df/tbl/data.frame)
## $ summit : num [1:849] 5710 5750 5715 5760 5675 ...
## $ debris_cov : num [1:849] 0 0 0 0 0 0 0 0 0 0 ...
## $ morph_type : num [1:849] 2 2 2 3 2 2 2 2 2 2 ...
## $ length : num [1:849, 1] -0.386 -0.284 -0.548 -0.579 0.295 ...
## ..- attr(*, "scaled:center")= num 6216
## ..- attr(*, "scaled:scale")= num 5539
## $ area : num [1:849, 1] -0.3439 -0.2619 -0.4021 -0.4901 0.0947 ...
## ..- attr(*, "scaled:center")= num 1393269
## ..- attr(*, "scaled:scale")= num 2117729
## $ P_snow : num [1:849, 1] -0.5453 -0.2486 -0.0552 0.9086 0.1799 ...
## ..- attr(*, "scaled:center")= num 120
## ..- attr(*, "scaled:scale")= num 50.7
## $ P_year : num [1:849, 1] -1.211 -0.811 -0.735 -0.782 -0.027 ...
## ..- attr(*, "scaled:center")= num 889
## ..- attr(*, "scaled:scale")= num 177
## $ P_monsoon : num [1:849, 1] -1.121 -0.593 -0.599 -0.564 0.481 ...
## ..- attr(*, "scaled:center")= num 638
## ..- attr(*, "scaled:scale")= num 94.4
## $ P_not_monsoon : num [1:849, 1] -1.162 -0.936 -0.788 -0.912 -0.538 ...
## ..- attr(*, "scaled:center")= num 251
## ..- attr(*, "scaled:scale")= num 93.3
## $ T_MIN_mean_monsoon : num [1:849, 1] -1.18 -1.59 -1.41 -1.72 -1.43 ...
## ..- attr(*, "scaled:center")= num 11.3
## ..- attr(*, "scaled:scale")= num 10.8
## $ T_MIN_mean_not_monsoon: num [1:849, 1] -1.24 -1.61 -1.43 -1.7 -1.47 ...
## ..- attr(*, "scaled:center")= num -119
## ..- attr(*, "scaled:scale")= num 13
## $ T_MAX_mean_monsoon : num [1:849, 1] -1.08 -1.59 -1.38 -1.75 -1.36 ...
## ..- attr(*, "scaled:center")= num 78.1
## ..- attr(*, "scaled:scale")= num 8.92
## $ T_MAX_mean_not_monsoon: num [1:849, 1] -1.16 -1.62 -1.41 -1.76 -1.46 ...
## ..- attr(*, "scaled:center")= num -27.5
## ..- attr(*, "scaled:scale")= num 10.4
## $ T_mean_mea.yr : num [1:849, 1] -1.18 -1.61 -1.42 -1.74 -1.45 ...
## ..- attr(*, "scaled:center")= num -24
## ..- attr(*, "scaled:scale")= num 10.8
## $ T_mean_monsoon : num [1:849, 1] -1.13 -1.6 -1.41 -1.74 -1.4 ...
## ..- attr(*, "scaled:center")= num 44.7
## ..- attr(*, "scaled:scale")= num 9.83
## $ T_mean_not_monsoon : num [1:849, 1] -1.21 -1.61 -1.43 -1.74 -1.47 ...
## ..- attr(*, "scaled:center")= num -73.1
## ..- attr(*, "scaled:scale")= num 11.6
## $ Slope_min : num [1:849, 1] 0.69002 -0.16035 0.41288 0.00442 -0.79263 ...
## ..- attr(*, "scaled:center")= num 8.45
## ..- attr(*, "scaled:scale")= num 6.45
## $ Slope_max : num [1:849, 1] -0.138 -0.748 -0.6 -0.855 -1.09 ...
## ..- attr(*, "scaled:center")= num 46.2
## ..- attr(*, "scaled:scale")= num 10.4
## $ Slope_mean : num [1:849, 1] -0.359 -0.925 -0.342 -0.431 -1.285 ...
## ..- attr(*, "scaled:center")= num 27.5
## ..- attr(*, "scaled:scale")= num 6.44
## $ Elev_min : num [1:849, 1] 0.837 0.901 1.071 1.336 0.896 ...
## ..- attr(*, "scaled:center")= num 4800
## ..- attr(*, "scaled:scale")= num 407
## $ Elev_max : num [1:849, 1] 0.641 0.86 0.699 0.991 0.548 ...
## ..- attr(*, "scaled:center")= num 5505
## ..- attr(*, "scaled:scale")= num 292
## $ Elev_mean : num [1:849, 1] 0.994 1.377 1.464 1.71 1.191 ...
## ..- attr(*, "scaled:center")= num 5155
## ..- attr(*, "scaled:scale")= num 221
## $ T_variability : num [1:849, 1] -0.929 -1.527 -1.286 -1.73 -1.291 ...
## ..- attr(*, "scaled:center")= num 104
## ..- attr(*, "scaled:scale")= num 7.82
## - attr(*, "na.action")= 'exclude' Named int [1:4] 731 739 793 829
## ..- attr(*, "names")= chr [1:4] "731" "739" "793" "829"# plot of pc1 and pc2
res.pca <- PCA(pca[,c(1:22)], scale.unit = TRUE, ncp = 5, graph = TRUE) 
# based on this PCA we can intially see that elev mean is closly aligned with PC1 and has a strong negative orientation, while being diamertically opposed to the agregated temperature and precipitation measures. Geo-physical conditions (aside from elev mean on PC1) tend to be ordinated on PC2 with the strongers factors being glacier area, length, and summit height/max elevation (the same redundent metric...).
#plot of pc1 and pc3
res.pca <- PCA(pca[,c(1:22)], scale.unit = TRUE, ncp = 5, graph = TRUE, axes = c(1,3))
# based on this plot of PC1 and PC2, we can see that snow precipitation is the main variable contributing to PC3s explained variation. Furthermore the ordination of variables on PC1 are only slightly altered in compairoson to the plot of PC1 and PC2.
pca <- prcomp(pca, scale. = T)
summary(pca)## Importance of components:
## PC1 PC2 PC3 PC4 PC5 PC6 PC7
## Standard deviation 3.239 2.1197 1.7267 1.26260 1.03151 0.82279 0.68121
## Proportion of Variance 0.456 0.1954 0.1296 0.06931 0.04626 0.02943 0.02018
## Cumulative Proportion 0.456 0.6514 0.7810 0.85032 0.89658 0.92601 0.94619
## PC8 PC9 PC10 PC11 PC12 PC13 PC14
## Standard deviation 0.57066 0.52564 0.41415 0.37528 0.32194 0.28148 0.25031
## Proportion of Variance 0.01416 0.01201 0.00746 0.00612 0.00451 0.00344 0.00272
## Cumulative Proportion 0.96035 0.97236 0.97982 0.98594 0.99045 0.99389 0.99662
## PC15 PC16 PC17 PC18 PC19 PC20
## Standard deviation 0.20272 0.15590 0.11041 0.01307 0.006805 0.004639
## Proportion of Variance 0.00179 0.00106 0.00053 0.00001 0.000000 0.000000
## Cumulative Proportion 0.99840 0.99946 0.99999 1.00000 1.000000 1.000000
## PC21 PC22 PC23
## Standard deviation 9.62e-08 9.232e-16 5.654e-16
## Proportion of Variance 0.00e+00 0.000e+00 0.000e+00
## Cumulative Proportion 1.00e+00 1.000e+00 1.000e+00pca$rotation[,1:6]## PC1 PC2 PC3 PC4
## summit -0.1156595114 -0.396530022 0.006722996 -0.1575511536
## debris_cov 0.0592119096 -0.269462680 -0.052365271 0.0841236946
## morph_type 0.0400162434 -0.091990076 -0.029584626 0.0998727770
## length 0.0483474510 -0.419110439 0.066102380 0.2009610901
## area 0.0498102447 -0.403993717 0.069942759 0.2063360677
## P_snow 0.0092334579 0.044215327 0.512080560 -0.1067217199
## P_year 0.1663450945 0.026036585 0.473270085 -0.0983638200
## P_monsoon 0.1149693609 -0.015678340 0.482582772 -0.1269818451
## P_not_monsoon 0.1989915206 0.065216422 0.408839901 -0.0579761041
## T_MIN_mean_monsoon 0.3060541207 0.012778292 -0.053013928 0.0008543518
## T_MIN_mean_not_monsoon 0.3032010943 -0.016791648 -0.044913670 -0.0236876350
## T_MAX_mean_monsoon 0.2981733398 0.062127235 -0.096162241 0.0460564355
## T_MAX_mean_not_monsoon 0.3038968050 0.008468283 -0.074287604 0.0015814723
## T_mean_mea.yr 0.3054852777 0.009768718 -0.064004869 0.0001986578
## T_mean_monsoon 0.3041721614 0.035272945 -0.072790921 0.0211532886
## T_mean_not_monsoon 0.3047792037 -0.005670976 -0.058388518 -0.0124484181
## Slope_min -0.0008673191 0.172349229 -0.144086417 -0.5497905945
## Slope_max 0.1074204433 -0.340938679 -0.052861818 -0.2748845202
## Slope_mean 0.0969690782 -0.129444291 -0.151594973 -0.6427029063
## Elev_min -0.2245285603 0.284072988 0.023014192 -0.0242427371
## Elev_max -0.1272311906 -0.398418302 0.024459017 -0.1469850576
## Elev_mean -0.2903997157 -0.014890301 0.052015325 -0.1132050921
## T_variability 0.2804581157 0.087127203 -0.134391044 0.0736330210
## PC5 PC6
## summit 0.0824178265 -0.013314116
## debris_cov -0.1631885443 -0.921362046
## morph_type -0.8875920506 0.231071412
## length -0.0376147737 0.144502015
## area -0.1262642924 0.122834120
## P_snow -0.1045303257 -0.040817253
## P_year -0.0006254728 -0.028488009
## P_monsoon 0.0538899676 -0.034703229
## P_not_monsoon -0.0557100478 -0.018888474
## T_MIN_mean_monsoon 0.0092062920 -0.004116058
## T_MIN_mean_not_monsoon 0.0024765380 0.003884554
## T_MAX_mean_monsoon 0.0178121599 -0.010498486
## T_MAX_mean_not_monsoon 0.0277241682 -0.001477082
## T_mean_mea.yr 0.0136610122 -0.001637198
## T_mean_monsoon 0.0131736061 -0.006805675
## T_mean_not_monsoon 0.0138881375 0.001490011
## Slope_min -0.3032269288 -0.114844999
## Slope_max 0.1929150277 0.181647499
## Slope_mean -0.0472596556 0.047772626
## Elev_min -0.0550247898 -0.073168174
## Elev_max 0.0741599741 0.021785456
## Elev_mean -0.0209235332 -0.053078593
## T_variability 0.0487193561 -0.014295989# based on this roation, I will make the following assumption for which variables are analogs to each PC
# PC1: All tempurature variables (both monsoon & non-monsoon), tempurature variability, and elevation mean
# PC2: Geophysical conditions such as min/max elevation, debris cover, catchment length and area, and max slope
# PC3: All precipitation variables
# PC4: Slope, and to a far lesser degree catchment area, length and elevation
# PC5: Morphology type
var_exp <- data.frame(pc = c(1:23),
var_exp = pca$sdev^2 / sum(pca$sdev^2))
# add the variances cumulatively
var_exp$var_exp_cumsum <- cumsum(var_exp$var_exp)
var_exp## pc var_exp var_exp_cumsum
## 1 1 4.560124e-01 0.4560124
## 2 2 1.953597e-01 0.6513721
## 3 3 1.296334e-01 0.7810055
## 4 4 6.931112e-02 0.8503167
## 5 5 4.626144e-02 0.8965781
## 6 6 2.943396e-02 0.9260121
## 7 7 2.017606e-02 0.9461881
## 8 8 1.415870e-02 0.9603468
## 9 9 1.201310e-02 0.9723599
## 10 10 7.457546e-03 0.9798175
## 11 11 6.123134e-03 0.9859406
## 12 12 4.506432e-03 0.9904470
## 13 13 3.444813e-03 0.9938918
## 14 14 2.724236e-03 0.9966161
## 15 15 1.786818e-03 0.9984029
## 16 16 1.056764e-03 0.9994597
## 17 17 5.299700e-04 0.9999896
## 18 18 7.426372e-06 0.9999971
## 19 19 2.013656e-06 0.9999991
## 20 20 9.358421e-07 1.0000000
## 21 21 4.023442e-16 1.0000000
## 22 22 3.705982e-32 1.0000000
## 23 23 1.389756e-32 1.0000000ggplot(var_exp) +
geom_bar(aes(x = pc, y = var_exp), stat = "identity") +
geom_line(aes(x = pc, y = var_exp_cumsum)) +
theme_classic()
# Following the Elbow method, it seems like 3-5 PCs seems like an approapriate cutoff point. Despite being more difficult to interpret the final results, I decided to include 5 PCs, due to PCs 4 and 5 increasing the overall variance explained from ~80% to 90%. This also helps to ensure that the large number of variables in the model are captured in the PCs, especially as morphology type almost exclusively loads on PC5.
# scores
pca_scores <- data.frame(pca$x)
head(pca_scores)## PC1 PC2 PC3 PC4 PC5 PC6
## 1 -4.085950 0.12736785 -1.15488972 -0.4276191 0.8461927 -0.19220332
## 2 -5.283278 -0.02000121 0.10140922 0.2929743 0.9265436 -0.25750181
## 3 -4.742488 0.34258840 -0.03086408 -0.5009767 0.7503837 -0.36280031
## 4 -5.739056 0.13114428 0.67963939 -0.2685366 -0.6152369 -0.08876674
## 5 -4.418274 -0.03063813 1.46720119 1.0095502 0.9809866 -0.22197371
## 6 -3.354310 0.37954311 0.86186526 0.4488272 1.1837522 -0.16445739
## PC7 PC8 PC9 PC10 PC11 PC12
## 1 -0.5924383 0.3157648 -0.06914327 0.040799740 -0.33150857 0.003904768
## 2 -0.4260974 -0.0194212 0.07569821 -0.284495967 -0.08137097 -0.125315391
## 3 -0.4281015 0.1150437 0.10357952 0.006878345 0.01024527 -0.130232249
## 4 0.3062982 0.4004047 0.59410478 -0.129178425 0.05062106 0.089638527
## 5 -0.4874418 -0.4850412 -0.13692691 -0.028121838 0.24060972 -0.082023890
## 6 0.2822511 -0.2910021 -0.38920635 -0.053532077 0.18451985 -0.127911367
## PC13 PC14 PC15 PC16 PC17 PC18
## 1 -0.06771404 -0.007749453 0.03161404 -0.08157295 -0.02674952 -0.02366540
## 2 -0.12582001 0.104398650 0.04780135 -0.07585605 -0.01790055 -0.01942436
## 3 -0.16104792 0.015764018 0.06123426 -0.03189586 -0.02538805 -0.01376238
## 4 0.16573447 -0.050168883 0.09239933 -0.16075389 -0.01365135 -0.02749751
## 5 0.19389144 0.139088085 -0.01502534 -0.08903185 0.07761670 -0.01844080
## 6 0.31287542 -0.240958925 -0.19863270 0.03489515 0.11186253 -0.01939790
## PC19 PC20 PC21 PC22 PC23
## 1 0.008769504 0.0026651323 -2.861286e-08 2.775558e-16 -2.255141e-16
## 2 0.001274638 0.0082995976 7.317158e-08 8.881784e-16 -4.024558e-16
## 3 -0.002945277 0.0004759622 -4.582566e-08 -1.110223e-16 -4.614364e-16
## 4 0.001010093 -0.0047553419 2.704832e-07 2.220446e-16 -2.636780e-16
## 5 0.003278927 -0.0033848377 1.013831e-07 1.665335e-15 -2.706169e-16
## 6 0.003915545 -0.0039411928 -2.274929e-07 -5.551115e-17 1.110223e-16pca_scores_df <- cbind(glacier_scaled, pca_scores)
#ggpairs(pca_scores_df[,c(4:6,8:9,22:25,38:39,52:53,66:75,76:80)]) # first 5 Pcs
m_pca <- glm(ELA ~ PC1 + PC2 + PC3 + PC4 + PC5, data = pca_scores_df)
summary(m_pca)##
## Call:
## glm(formula = ELA ~ PC1 + PC2 + PC3 + PC4 + PC5, data = pca_scores_df)
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -365.98 -41.14 -3.62 41.31 562.01
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 5104.9305 2.8600 1784.941 < 2e-16 ***
## PC1 -65.1737 0.8836 -73.757 < 2e-16 ***
## PC2 17.7810 1.3500 13.171 < 2e-16 ***
## PC3 4.7192 1.6573 2.848 0.00451 **
## PC4 -24.0376 2.2665 -10.606 < 2e-16 ***
## PC5 -4.2597 2.7743 -1.535 0.12506
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for gaussian family taken to be 6944.475)
##
## Null deviance: 45691145 on 848 degrees of freedom
## Residual deviance: 5854193 on 843 degrees of freedom
## AIC: 9927.3
##
## Number of Fisher Scoring iterations: 2# Here we see that PCs 1,2, and 4 have the strongest effect size and large predictive power for ELA. Based on the caracterization of which variables are represented by each PC, this means that tempurature, geophysical conditions, and slope (and to a smaller extend catchment length, area, and elevation) are the most important variables. Out of the three PCs, PC1 (tempurature variables), are by far the most important with a effect size of -65.17 (Std. error 0.88). PCs 2 (geophysical conditions) and 4 (slope) have effect sizes of 17.78 (Std. error 1.35) and -24.04 (Std. error 2.27) respectively.
# R squared
r.squaredGLMM(m_pca)## R2m R2c
## [1,] 0.8712126 0.8712126# The R² is 87.12 %
plot(m_pca)
# the residual vs fitted plot shows a mostly flat and even distribution of points, with some outlying point at intermidiate values. The QQ-plot has skewed tails at both the lower and upper bounds and is substaintial for larger values
dwplot(m_pca) +
geom_vline(xintercept = 0, colour = "grey60", linetype = 2)
# dELA model
m_pca_d <- glm(dELA ~ PC1 + PC2 + PC3 + PC4 + PC5, family = "poisson", data = pca_scores_df)
summary(m_pca)##
## Call:
## glm(formula = ELA ~ PC1 + PC2 + PC3 + PC4 + PC5, data = pca_scores_df)
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -365.98 -41.14 -3.62 41.31 562.01
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 5104.9305 2.8600 1784.941 < 2e-16 ***
## PC1 -65.1737 0.8836 -73.757 < 2e-16 ***
## PC2 17.7810 1.3500 13.171 < 2e-16 ***
## PC3 4.7192 1.6573 2.848 0.00451 **
## PC4 -24.0376 2.2665 -10.606 < 2e-16 ***
## PC5 -4.2597 2.7743 -1.535 0.12506
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for gaussian family taken to be 6944.475)
##
## Null deviance: 45691145 on 848 degrees of freedom
## Residual deviance: 5854193 on 843 degrees of freedom
## AIC: 9927.3
##
## Number of Fisher Scoring iterations: 2# R squared
r.squaredGLMM(m_pca_d)## Warning: The null model is correct only if all variables used by the original
## model remain unchanged.## R2m R2c
## delta 0.7644043 0.7644043
## lognormal 0.7652960 0.7652960
## trigamma 0.7635058 0.7635058# The R² is 76.44 %
plot(m_pca_d)
# the residual vs fitted plot shows a mostly flat and even distribution of points, with some outlying point at intermidiate values
dwplot(m_pca_d) +
geom_vline(xintercept = 0, colour = "grey60", linetype = 2)
Automated variable selection
For the automated variable variable I focused on variables that I previously found to be strong predictors of ELA, aswell as the de-agrigated forms of temperature and precipitation metrics. While potential interaction between temperature and precipitation variables are likely to exist, these were not included in the saturated model, as just including a reduced list of variables with no interactions, already brought my computer to the edges of its computational limits.
# create semi-saturated model, with the broader variables based on my novice guess. These variables where selected based on a priori assumptions on what I thought would be will be more relavent to include, based on the outputs of my previous models and exploratory analysis
m_saturated <-
glm( ELA ~ debris_cov + morph_type + orientation + length + P_snow + P_year + P_not_monsoon + T_mean_mea.yr + T_mean_monsoon + T_mean_not_monsoon + Slope_min + Elev_min + Elev_max + Elev_mean, # + T_MIN_mean_monsoon + T_MIN_mean_not_monsoon + T_MAX_mean_monsoon T_MAX_mean_not_monsoon ; taken out due to computational constraints. Additionally, to get the model to run, area, max/mean slope, t_variability and monsoon precipition also needed to be removed. I chose these variables, as in previous models and in explotatory analysis they tended have low impacts on ELA.
data = glacier_scaled,
family = "gaussian")
summary(m_saturated)##
## Call:
## glm(formula = ELA ~ debris_cov + morph_type + orientation + length +
## P_snow + P_year + P_not_monsoon + T_mean_mea.yr + T_mean_monsoon +
## T_mean_not_monsoon + Slope_min + Elev_min + Elev_max + Elev_mean,
## family = "gaussian", data = glacier_scaled)
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -319.00 -22.28 -0.55 21.49 273.08
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 5.087e+03 8.798e+00 578.204 < 2e-16 ***
## debris_cov 1.875e+01 7.928e+00 2.365 0.018255 *
## morph_type -2.507e-01 2.733e+00 -0.092 0.926953
## orientationN 1.495e+01 5.812e+00 2.572 0.010297 *
## orientationNE 1.537e+01 6.553e+00 2.345 0.019255 *
## orientationNW 2.370e+01 7.226e+00 3.280 0.001081 **
## orientationS 2.711e+01 6.794e+00 3.991 7.17e-05 ***
## orientationSE 6.481e+00 8.088e+00 0.801 0.423155
## orientationSW 2.608e+01 7.460e+00 3.496 0.000498 ***
## orientationW 2.775e+01 6.691e+00 4.147 3.72e-05 ***
## length -8.992e+00 3.391e+00 -2.652 0.008158 **
## P_snow 6.372e+00 3.453e+00 1.845 0.065359 .
## P_year 1.664e+01 6.409e+00 2.597 0.009567 **
## P_not_monsoon -3.239e+01 7.727e+00 -4.192 3.06e-05 ***
## T_mean_mea.yr -4.231e+05 1.411e+07 -0.030 0.976093
## T_mean_monsoon 1.597e+05 5.329e+06 0.030 0.976091
## T_mean_not_monsoon 2.646e+05 8.830e+06 0.030 0.976097
## Slope_min 5.444e+00 1.985e+00 2.742 0.006237 **
## Elev_min 1.803e+02 5.926e+00 30.421 < 2e-16 ***
## Elev_max 8.835e+01 4.591e+00 19.244 < 2e-16 ***
## Elev_mean 6.351e+00 7.194e+00 0.883 0.377567
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for gaussian family taken to be 2401.556)
##
## Null deviance: 45691145 on 848 degrees of freedom
## Residual deviance: 1988489 on 828 degrees of freedom
## AIC: 9040.6
##
## Number of Fisher Scoring iterations: 2options(na.action = "na.fail") # Required for dredge to run
# Run all possible model permutations and tank according to AIC values
m_dredge <- dredge(m_saturated, beta = F, evaluate = T, rank = AICc)## Fixed term is "(Intercept)"options(na.action = "na.omit") # set back to default
head(m_dredge)## Global model call: glm(formula = ELA ~ debris_cov + morph_type + orientation + length +
## P_snow + P_year + P_not_monsoon + T_mean_mea.yr + T_mean_monsoon +
## T_mean_not_monsoon + Slope_min + Elev_min + Elev_max + Elev_mean,
## family = "gaussian", data = glacier_scaled)
## ---
## Model selection table
## (Int) dbr_cov Elv_max Elv_men Elv_min lng orn P_not_mns P_snw P_yer
## 10204 5086 19.26 90.23 183.9 -9.266 + -30.19 5.733 15.09
## 9948 5086 19.37 91.32 184.9 -9.292 + -24.51 15.51
## 8156 5086 19.04 90.97 183.6 -9.328 + -32.47 6.361 16.73
## 14300 5086 19.04 90.97 183.6 -9.328 + -32.47 6.361 16.73
## 12252 5086 19.04 90.97 183.6 -9.328 + -32.47 6.361 16.73
## 10208 5086 19.03 88.16 4.689 181.4 -9.070 + -29.66 5.595 14.72
## Slp_min T_men_mea.yr T_men_mns T_men_not_mns df logLik AICc delta
## 10204 5.427 -28.51 18 -4498.965 9034.8 0.00
## 9948 5.395 -30.98 17 -4500.478 9035.7 0.94
## 8156 5.498 -60.91 33.41 19 -4498.702 9036.3 1.57
## 14300 5.498 10.41 -38.10 19 -4498.702 9036.3 1.57
## 12252 5.498 27.58 -55.36 19 -4498.702 9036.3 1.57
## 10208 5.348 -26.91 19 -4498.733 9036.4 1.63
## weight
## 10204 0.291
## 9948 0.182
## 8156 0.133
## 14300 0.133
## 12252 0.133
## 10208 0.129
## Models ranked by AICc(x)nrow(m_dredge) # 16384 models in total## [1] 16384top_model <- get.models(m_dredge, subset = 1)[[1]]
top_model##
## Call: glm(formula = ELA ~ debris_cov + Elev_max + Elev_min + length +
## orientation + P_not_monsoon + P_snow + P_year + Slope_min +
## T_mean_not_monsoon + 1, family = "gaussian", data = glacier_scaled)
##
## Coefficients:
## (Intercept) debris_cov Elev_max Elev_min
## 5086.266 19.258 90.232 183.873
## length orientationN orientationNE orientationNW
## -9.266 14.967 15.421 23.393
## orientationS orientationSE orientationSW orientationW
## 27.646 7.058 26.234 27.895
## P_not_monsoon P_snow P_year Slope_min
## -30.188 5.733 15.093 5.427
## T_mean_not_monsoon
## -28.514
##
## Degrees of Freedom: 848 Total (i.e. Null); 832 Residual
## Null Deviance: 45690000
## Residual Deviance: 1992000 AIC: 9034# Summarize top model
summary(top_model)##
## Call:
## glm(formula = ELA ~ debris_cov + Elev_max + Elev_min + length +
## orientation + P_not_monsoon + P_snow + P_year + Slope_min +
## T_mean_not_monsoon + 1, family = "gaussian", data = glacier_scaled)
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -319.02 -22.06 -0.35 21.54 274.51
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 5086.266 4.483 1134.445 < 2e-16 ***
## debris_cov 19.258 7.854 2.452 0.014408 *
## Elev_max 90.232 3.345 26.971 < 2e-16 ***
## Elev_min 183.873 4.494 40.917 < 2e-16 ***
## length -9.266 3.227 -2.871 0.004192 **
## orientationN 14.967 5.797 2.582 0.009994 **
## orientationNE 15.421 6.537 2.359 0.018550 *
## orientationNW 23.393 7.206 3.246 0.001216 **
## orientationS 27.646 6.726 4.110 4.34e-05 ***
## orientationSE 7.058 8.052 0.877 0.381004
## orientationSW 26.234 7.412 3.539 0.000424 ***
## orientationW 27.895 6.616 4.216 2.76e-05 ***
## P_not_monsoon -30.188 6.948 -4.345 1.57e-05 ***
## P_snow 5.733 3.327 1.723 0.085203 .
## P_year 15.093 5.881 2.566 0.010454 *
## Slope_min 5.427 1.950 2.783 0.005505 **
## T_mean_not_monsoon -28.514 4.442 -6.419 2.31e-10 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for gaussian family taken to be 2393.755)
##
## Null deviance: 45691145 on 848 degrees of freedom
## Residual deviance: 1991604 on 832 degrees of freedom
## AIC: 9033.9
##
## Number of Fisher Scoring iterations: 2# psudo R^2
r.squaredGLMM(top_model) # 95.74%## R2m R2c
## [1,] 0.9556106 0.9556106plot(top_model)
# the residual vs fitted plot shows a mostly flat and even distribution of points, with greater diviances seen at the lower spectrum of points, likely due to the negative skew described above
qqnorm(resid(top_model))
qqline(resid(top_model))
# When checking the QQ-plot it is seen that there is a left tail at lower bonds and a right tail at the upper bonds. Following an ad hoc transformation of the dependant variable ELA, this observed trend only showed a very marginal improvment, so I decided to keep ELA in its untransformed state.
dwplot(top_model) +
geom_vline(xintercept = 0, colour = "grey60", linetype = 2)
# The automated model section based on AIC ranking of 16384 models, indicated that min and max elevation have are the two most important variables to predict ELA(with effect sizes 183.87 and 90.23 respectily & with small std.errors of 4.49 and 3.34), with large effect sizes relative to other variables. The other next most important variables where the various orientation directions, aswell as preciption and temperature in non monsoon periods. A noteable ommison is mean yearly temperature, which in the glmer model had the highest effect size. This could be attributed to high colinearity between mean temperature and temperature in non monsoon periods, but with temperature in non monsoon periods have a large effect size and therefor being prioritized in the selection. Q2. Can we explain ELA changes since the Little Ice Age with topographical and climatological variables?
m_saturated_d <-
glm( dELA ~ debris_cov + morph_type + orientation + length + P_year + P_not_monsoon + T_mean_mea.yr + T_mean_monsoon + T_mean_not_monsoon + Slope_min + Slope_mean + Elev_min + Elev_max + Elev_mean, # + T_MIN_mean_monsoon + T_MIN_mean_not_monsoon + T_MAX_mean_monsoon T_MAX_mean_not_monsoon ; taken out due to computational constraints. Additionally, to get the model to run, area, max slope, t_variability snow, and monsoon precipition also needed to be removed. I chose these variables, as in previous models and in explotatory analysis they tended have low impacts on ELA.
data = glacier_scaled,
family = "gaussian")
summary(m_saturated_d)##
## Call:
## glm(formula = dELA ~ debris_cov + morph_type + orientation +
## length + P_year + P_not_monsoon + T_mean_mea.yr + T_mean_monsoon +
## T_mean_not_monsoon + Slope_min + Slope_mean + Elev_min +
## Elev_max + Elev_mean, family = "gaussian", data = glacier_scaled)
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -117.94 -34.45 -6.80 27.19 337.05
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.027e+02 9.506e+00 10.807 < 2e-16 ***
## debris_cov -2.581e+01 8.558e+00 -3.015 0.002645 **
## morph_type -2.464e+00 2.954e+00 -0.834 0.404599
## orientationN -3.077e+00 6.298e+00 -0.489 0.625318
## orientationNE -1.553e+00 7.068e+00 -0.220 0.826119
## orientationNW -1.135e+00 7.819e+00 -0.145 0.884588
## orientationS 2.305e+01 7.359e+00 3.133 0.001794 **
## orientationSE 4.858e+00 8.729e+00 0.557 0.577990
## orientationSW 2.228e+01 8.080e+00 2.758 0.005948 **
## orientationW 1.725e+01 7.216e+00 2.390 0.017068 *
## length 5.503e+00 3.808e+00 1.445 0.148752
## P_year 6.092e+00 6.936e+00 0.878 0.380065
## P_not_monsoon -7.430e+00 7.174e+00 -1.036 0.300616
## T_mean_mea.yr 7.542e+06 1.520e+07 0.496 0.619925
## T_mean_monsoon -2.847e+06 5.739e+06 -0.496 0.619923
## T_mean_not_monsoon -4.718e+06 9.510e+06 -0.496 0.619928
## Slope_min -3.628e+00 2.484e+00 -1.461 0.144496
## Slope_mean 1.233e+01 2.911e+00 4.236 2.53e-05 ***
## Elev_min 2.941e+01 6.422e+00 4.580 5.37e-06 ***
## Elev_max 1.907e+01 5.151e+00 3.702 0.000228 ***
## Elev_mean -1.581e+01 7.868e+00 -2.010 0.044805 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for gaussian family taken to be 2796.718)
##
## Null deviance: 2869609 on 848 degrees of freedom
## Residual deviance: 2315682 on 828 degrees of freedom
## AIC: 9169.9
##
## Number of Fisher Scoring iterations: 2options(na.action = "na.fail") # Required for dredge to run
# Run all possible model permutations and tank according to AIC values
model_dredge_d <- dredge(m_saturated_d, beta = F, evaluate = T, rank = AICc)## Fixed term is "(Intercept)"options(na.action = "na.omit") # set back to default
head(model_dredge_d)## Global model call: glm(formula = dELA ~ debris_cov + morph_type + orientation +
## length + P_year + P_not_monsoon + T_mean_mea.yr + T_mean_monsoon +
## T_mean_not_monsoon + Slope_min + Slope_mean + Elev_min +
## Elev_max + Elev_mean, family = "gaussian", data = glacier_scaled)
## ---
## Model selection table
## (Int) dbr_cov Elv_max Elv_men Elv_min lng orn Slp_men Slp_min
## 7760 96.29 -24.74 22.42 -16.50 27.21 + 11.41 -4.532
## 13904 96.29 -24.74 22.42 -16.50 27.21 + 11.41 -4.532
## 11856 96.29 -24.74 22.42 -16.50 27.21 + 11.41 -4.532
## 7776 96.03 -26.20 19.95 -16.09 29.74 4.846 + 12.45 -4.007
## 13920 96.03 -26.20 19.95 -16.09 29.74 4.846 + 12.45 -4.007
## 11872 96.03 -26.20 19.95 -16.09 29.74 4.846 + 12.45 -4.007
## T_men_mea.yr T_men_mns T_men_not_mns df logLik AICc delta weight
## 7760 75.91 -52.35 17 -4565.181 9165.1 0.00 0.177
## 13904 -23.69 47.49 17 -4565.181 9165.1 0.00 0.177
## 11856 -62.75 86.74 17 -4565.181 9165.1 0.00 0.177
## 7776 74.48 -50.60 18 -4564.261 9165.3 0.25 0.156
## 13920 -22.48 46.59 18 -4564.261 9165.3 0.25 0.156
## 11872 -59.56 83.85 18 -4564.261 9165.3 0.25 0.156
## Models ranked by AICc(x)nrow(model_dredge_d) # 16384 models in total## [1] 16384top_model_d <- get.models(model_dredge_d, subset = 1)[[1]]
top_model_d##
## Call: glm(formula = dELA ~ debris_cov + Elev_max + Elev_mean + Elev_min +
## orientation + Slope_mean + Slope_min + T_mean_mea.yr + T_mean_monsoon +
## 1, family = "gaussian", data = glacier_scaled)
##
## Coefficients:
## (Intercept) debris_cov Elev_max Elev_mean Elev_min
## 96.293 -24.740 22.417 -16.498 27.209
## orientationN orientationNE orientationNW orientationS orientationSE
## -3.414 -2.425 -1.927 23.096 4.273
## orientationSW orientationW Slope_mean Slope_min T_mean_mea.yr
## 21.730 16.140 11.413 -4.532 75.911
## T_mean_monsoon
## -52.346
##
## Degrees of Freedom: 848 Total (i.e. Null); 833 Residual
## Null Deviance: 2870000
## Residual Deviance: 2328000 AIC: 9164# Summarize top model
summary(top_model_d)##
## Call:
## glm(formula = dELA ~ debris_cov + Elev_max + Elev_mean + Elev_min +
## orientation + Slope_mean + Slope_min + T_mean_mea.yr + T_mean_monsoon +
## 1, family = "gaussian", data = glacier_scaled)
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -114.17 -35.46 -8.50 27.54 335.72
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 96.293 4.841 19.891 < 2e-16 ***
## debris_cov -24.740 8.383 -2.951 0.00325 **
## Elev_max 22.417 4.702 4.768 2.20e-06 ***
## Elev_mean -16.498 7.856 -2.100 0.03603 *
## Elev_min 27.209 6.091 4.467 9.03e-06 ***
## orientationN -3.414 6.140 -0.556 0.57838
## orientationNE -2.425 6.976 -0.348 0.72822
## orientationNW -1.927 7.791 -0.247 0.80473
## orientationS 23.096 7.209 3.204 0.00141 **
## orientationSE 4.273 8.691 0.492 0.62306
## orientationSW 21.730 7.900 2.751 0.00608 **
## orientationW 16.140 7.180 2.248 0.02484 *
## Slope_mean 11.413 2.788 4.094 4.65e-05 ***
## Slope_min -4.532 2.419 -1.873 0.06137 .
## T_mean_mea.yr 75.911 23.609 3.215 0.00135 **
## T_mean_monsoon -52.346 23.250 -2.251 0.02462 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for gaussian family taken to be 2794.487)
##
## Null deviance: 2869609 on 848 degrees of freedom
## Residual deviance: 2327807 on 833 degrees of freedom
## AIC: 9164.4
##
## Number of Fisher Scoring iterations: 2# psudo R^2
r.squaredGLMM(top_model_d) # 83.01%## R2m R2c
## [1,] 0.1860884 0.1860884plot(top_model_d)
# the residual vs fitted plot shows a mostly flat and even distribution of points, with greater diviances seen at the lower spectrum of points, likely due to the negative skew described above
# When checking the QQ-plot it is seen that there is a left tail at lower bonds and a right tail at the upper bonds. Following an ad hoc transformation of the dependant variable ELA, this observed trend only showed a very marginal improvment, so I decided to keep ELA in its untransformed state.
dwplot(top_model_d) +
geom_vline(xintercept = 0, colour = "grey60", linetype = 2)
Compairisons and conclusions
AICc_df <- AICc(m_lme4, m_lme4_d, m_bays, m_bays_d, m_pca, m_pca_d, top_model, top_model_d) %>%
arrange(AICc)
# Based on a compaired ranking between the modeles contructed under my three approaches, it is seen that automated variable selection performed best for the current ELA, while the lme4 had the lowrst AICc for dELA.